Wigner Functional Method
in
Quantum Field Theory
Berndt Müller
YIPQS Molecule: “Entropy Production”
YITP Kyoto - 13 August 2008
Based on: S. Mrówczyński & BM, PRD 50, 7542 (1994)
An impossible dream?
Simulations of real-time dynamics of quantum fields far off equilibrium have used a
variety of approximation schemes:

Boltzmann equations for (quasi-)particle excitations:


Classical nonlinear (lattice) field equations:


Good in dilute systems, but loss of coherence and off-shell effects.
Good for highly occupied modes, but loss of all quantum effects.
Real-time Green function techniques:

Practical for homogeneous or low-dimensional systems only.
Is it possible to devise a scheme which encompasses the classical field and particle
limit (useful for phenomenology), yet includes essential quantum effects, such as the
uncertainty relation, phases, or correlations, yet can be treated by similar methods
similar to those that have already been developed?
2
Wigner function
Reminder - Usual definition of Wigner function for a point particle:
!
1
1
−ipu
W (q, p; t) = du e
!q + u| ρ̂(t) |q − u#
2
2
Properties:
!
W (q, p) = W (q, p)∗
dqdp
W (q, p) = Tr ρ̂ = 1
2π
Equation of motion (for H = p²/m + V) :
∂W
∂t
=
!→0
−→
p
−
m
p
−
m
!
"
#
"
#$
∂W
i ∂V
i! ∂
∂V
i! ∂
−
q+
−
q−
W
∂q
! ∂q
2 ∂p
∂q
2 ∂p
∂W
∂V ∂W
+
∂q
∂q ∂p
3
Wave packets
!
iP q−∆(q−a)2
ΨP,a (q) = 2∆/π e
!
"
1
2
W (q, p) = 2 exp −2∆(q − a) −
(p − P )2
2∆
Consider a Gaussian wavepacket:
The Wigner transform is:
Pure quantum state:
Tr ρ̂ = Tr ρ̂ = 1
2
↔
1
!(q − a) # =
4∆
!(p − P )2 # = ∆
2
2 1/2
(!(∆q) "!(∆P ) "
2
!
dqdp
W (q, p)2 =
2π
!
dqdp
W (q, p) = 1
2π
Δ→∞: Good position
Δ→0: Good momentum
1
=
2
Minumum uncertainty wavepacket
4
Wigner functional
Adapt quantum phase space formulation (Wigner function) to the field representation
of quantum field theory in order make the classical field limit more transparent.
Wigner functional = Adaptation to QFT. Start with a scalar quantum field Φ.
Position space representation:
W [Φ(x), Π(x); t]
=
!
" !
#
1
1
Dϕ(x) exp −i dx Π(x)ϕ(x) "Φ(x) + ϕ(x)| ρ̂(t) |Φ(x) − ϕ(x)#
2
2
Momentum space representation:
W [Φ(p), Π(p); t] =
!
" !
Dϕ(p) exp −i
0
∞
%
$
1
1
dp Π∗ (p)ϕ(p) + Π(p)ϕ∗ (p) "Φ(p) + ϕ(p)| ρ̂(t) |Φ(p) − ϕ(p)#
2
2
#
5
Basics
!
Using
" !
#
DΠ
exp ±i dx Π(x)ϕ(x) = δ(ϕ(x))
2π
W [Φ(x), Π(x); t]
implies
=
!
it is obvious that
" !
#
1
1
Dϕ(x) exp −i dx Π(x)ϕ(x) "Φ(x) + ϕ(x)| ρ̂(t) |Φ(x) − ϕ(x)#
2
2
Z ≡ Tr ρ̂ =
!
DΦ(x)"Φ(x)| ρ̂ |Φ(x)# =
!
DΠ(x)
DΦ(x)
W [Φ, Π; t]
2π
Similarly, by inserting complete sets of states and partial integrations:
!
"
1
1
!O(Φ̂, Π̂)" =
Tr ρ̂(t)O(Φ̂, Π̂) =
Z
Z
#
DΠ(x)
DΦ(x)
O(Φ, Π) W [Φ, Π; t]
2π
Wigner functional expression corresponds to symmetrized quantum operators.
6
Equation of motion - I
1
Ĥ =
2
∂
i! ρ̂(t) = [Ĥ, ρ̂(t)]
∂t
!
"
%
#
$
2
dx Π̂2 (x) + ∇Φ̂(x) + m2 Φ̂2 (x) − L̂I (x)
"
#
!
∂
i
1
1
W [Φ, Π; t] =
Dϕ exp −
dx Π(x)ϕ(x) "Φ + ϕ| [Ĥ, ρ̂] |Φ − ϕ#
∂t
!
2
2
≡ GΠ + G∇ + Gm + G I
!
"
#
!
m
i
=
dx Dϕ exp −
dx Π(x)ϕ(x)
2
!
$%
'(
&2 %
&
)
1
2
1
1
1
×
Φ(x) + ϕ(x) − Φ(x) − 2 ϕ(x)
Φ + 2 ϕ| ρ̂ |Φ − 2 ϕ
2
2
Gm
!
!
Gm = i!m
2
!
δ
dx Φ(x)
W [Φ, Π; t]
δΠ(x)
7
Equation of motion - II
Other terms are derived similarly, final result is:
!
with
∂
+
∂t
"
dx
#
$ 2
% δ
δ
2
Π(x)
− m Φ(x) − ∇ Φ(x)
δΦ(x)
δΠ(x)
i
KI (x) ≡ − LI
!
!
i! δ
Φ(x) +
2 δΠ(x)
"
i
+ LI
!
!
&'
+ KI (x) W [Φ, Π; t] = 0
i! δ
Φ(x) −
2 δΠ(x)
"
Standard example: Φ⁴ theory:
λ 4
LI (Φ) = − Φ
4!
−→
λ
KI =
6
!
δ
!
δ
−Φ (x)
+ Φ(x) 3
δΠ(x)
4
δΠ (x)
2
3
3
"
Thus, evolution of W is described by a classical transport equation with quantum
corrections:
∂
W [Φ, Π; t] = −
∂t
!
dx
"
"
3
δH
δ
δH
δ
δ
−
+ O !2 3
δΠ(x) δΦ(x) δΦ(x) δΠ(x)
δΠ
##
W [Φ, Π; t]
8
Free field at T ≠ 0
Density
matrix:
1 −β Ĥ0
ρ̂ = e
Z
with
Ĥ0 =
!
∞
0
#
dp " †
Π̂ (p)Π̂(p) + (p2 + m2 )Φ̂† (p)Φ̂(p)
2π
Momentum space Wigner functional (ħ = 1):
"
&
# ∞
$
%
1
dp ˜
!
Wβ [Φ, Π] = C exp − β
∆β (p) Π∗ (p)Π(p) + E 2 (p)Φ∗ (p)Φ(p)
2 −∞ 2π
with
C ≡ exp
β→0:
!"
∞
−∞
dp
βE(p)
ln th
2π
2
˜ β (p) → 1.
∆
#
˜ β (p) =
∆
2
βE(p)
th
βE(p)
2
(classical limit)
Coordinate space form contains “nonlocal” Hamiltonian:
!
%
"
#
$
1
Wβ [Φ(x), Π(x)] = C ! exp − β dx dx! ∆(x − x! ) Π(x)Π(x! ) + ∇Φ(x)∇Φ(x! ) + m2 Φ(x)Φ(x! )
2
9
Fluctuations
Thermal quantum fluctuations are larger than classical fluctuations:
!Φ̂ (p)Φ̂(p)" =
†
!Π̂ (p)Π̂(p)" =
†
1
2 E(p) th βE(p)
2
E(p)
2
βE(p)
th 2
1
≥
2 E(p)
E(p)
≥
2
implying the uncertainty relation:
!
!Φ̂† (p)Φ̂(p)" !Π̂† (p)Π̂(p)" =
1
2 th βE(p)
2
1
≥
2
10
Generating functional
Perturbation theory is based on generating functional
" !
#
!
DΠ
Z[j(x)] ≡
DΦ
W [Φ, Π] exp β dx Φ(x)j(x)
2π
$
'
!
!
%
&
DΠ
=
DΦ
exp −β dx dx! H(x, x! ) − δ(x − x! )Φ(x)j(x! )
2π
Two point function:
Free field:
with
G(x) =
!
δ 2 Z[j] !!
1 1
!Φ̂(x)Φ̂(y)" = 2
!
β Z[j] δj(y)δj(x) !
!
β
Z0 [j(x)] = N exp
2
!
"
j(x)=0
#
dx dx! j(x)G(x − x! )j(x! )
1
dp
e−ipx
sh(2π|x|/β)
m→0
# −→
"
2
2
˜
2π ∆β (p) p + m
4π|x| ch(2π|x|/β) − 1
11
Particles
Consider one-particle state:
ρ̂ = a† (p)|0!"0|a(p)
Wigner functional:
!
"
!
"
2
2
2
|Π(p)| + E(p) Φ(p)|
|Π(p)| + E(p) Φ(p)|
W1 [Φ(p), Π(p)] = C 2
− 1 exp −
E(p)
E(p)
Partially occupied state:
2
2
2
ρ̂ = (1 − f (p))|0"#0| + f (p)a† (p)|0"#0|a(p)
Wigner functional:
!
"
#$
"
#
2
2
2
2
2
2
|Π(p)| + E(p) Φ(p)|
|Π(p)| + E(p) Φ(p)|
Wf [Φ(p), Π(p)] = C 1 + 2f (p)
− 1 exp −
E(p)
E(p)
"
#
2
2
2
|Π(p)|
+
E(p)
Φ(p)|
≈ C ! exp −
[1 + 2f (p)] E(p)
Agrees with thermal expression for f(p) = (e-βE(p)-1)-1.
12
Roll-over transition
Decay of an unstable “vacuum state” is a common problem e.g. in cosmology (end of
inflation) or condensed matter physics (phase transitions). Paradigm case: “roll-over”.
t<0
p2
m(t)2 2
Ĥ(t) =
+
x
2
2
|Ψ(x)|²
V(x)
|Ψ(x)|²
with
|Ψ(x)|²
V(x)
t=0
m2 (t) = m2 θ(−t) − µ2 θ(t)
|Ψ(x)|²
V(x)
t=1
V(x)
t=2
13
Wigner function
t=0
t = 0.5
t=1
t=2
p
x
14
Roll-over in QFT
Ĥ(t) =
!
0
∞
#
dp " †
Π̂ (p)Π̂(p) + (m2 (t) + p2 ) Φ̂† (p)Φ̂(p)
2π
m2 (t) = m2 θ(−t) − µ2 θ(t)
Initial state (not necessarily vacuum):
"
&
# ∞
$
%
1
dp
!
W [Φ, Π, t < 0] = C exp − β
∆β (p) Π∗ (p)Π(p) + E 2 (p)Φ∗ (p)Φ(p)
2 −∞ 2π
Hamiltonian for t > 0:
Ĥ(t > 0)
=
!
0
µ
# ! ∞ dp "
#
dp " †
2
†
†
2
†
Π̂ (p)Π̂(p) − ω− (p) Φ̂ (p)Φ̂(p) +
Π̂ (p)Π̂(p) + ω+ (p) Φ̂ (p)Φ̂(p)
2π
2π
µ
ω± (p) ≡
!
±(p2 − µ2 ) is always real
15
Solution
Classical field trajectories:
W [Φ(p), Π(p); t] = W [Φ(p, −t), Π(p, −t); 0]
d
2
Π(p, t) ± ω±
(p) Φ(p, t) = 0
dt
d
Φ(p, t) = Π(p, t)
dt
Φ(p, 0) = Φ(p),
with
Π(p, 0) = Π(p)
"
&
# ∞
$
%
1
dp ˜
!
W [Φ, Π; t] = C exp − β
∆β (p) Π∗ (p, −t)Π(p, −t) + E 2 (p)Φ∗ (p, −t)Φ(p, −t)
2 −∞ 2π
Φ(p, t)
Π(p, t)
Φ(p, t)
Π(p, t)
1
= Φ(p) ch(ω− (p)t) −
Π(p) sh(ω− (p)t)
ω− (p)
= Π(p) ch(ω− (p)t) − ω− (p) Φ(p) sh(ω− (p)t)
0<p<µ
1
Π(p) sin(ω+ (p)t)
ω+ (p)
= Π(p) cos(ω+ (p)t) + ω+ (p) Φ(p) sin(ω+ (p)t)
p>µ
= Φ(p) cos(ω+ (p)t) −
16
Field correlator
Distribution of field amplitudes:
! [Φ; t] ≡
W
"
DΠ !
2π W [Φ, Π; t]
#
= C ! exp − β2
χ(p, t)
with
"∞
0
= Θ(µ − p)
dp
2π
$
˜ β (p) χ(p, t) Φ∗ (p)Φ(p)
∆
2
E 2 (p)ω−
(p)
2 (p) ch2 (ω (p)t) + E 2 (p) sh2 (ω (p)t)
ω−
−
−
2
E 2 (p)ω+
(p)
+Θ(p − µ) 2
ω+ (p) cos2 (ω+ (p)t) + E 2 (p) sin2 (ω+ (p)t)
Field fluctuations:
!Φ̂ (p)Φ̂(p)"t = !Φ̂ (p)Φ̂(p)"0
†
†
!
"
E 2 (p) 2
sh (ω− (p)t)
ch (ω− (p)t) + 2
ω− (p)
2
Field correlation function:
G(x, t) =
!
−ipx
dp
e
˜ β (p) χ(p, t)
2π ∆
t→∞
−→
$
%
#
2
βmµ
µ
µx
−3/2
1 + 2 (πµt)
exp 2µt −
64 th (βm/2)
m
4t
"
2
17
Mean field approximation
Reduce nonlinear interactions by means of expectation values:
λ 4
λ 2
Φ̂ (x) → "Φ̂ (x)# Φ̂2 (x)
4!
4
m2∗
1
!Φ̂ (x)" =
Z
2
Self-consistent
equation for m*:
m2∗
!
λ 2
= m + !Φ̂ (x)"
2
2
DΠ(x) 2
DΦ(x)
Φ (x) W [Φ, Π; t]
2π
λ
= m + G(0)
2
"
! ∞
p2 + m2∗
λ
dp
1
2
= mren +
# "
$
2
2
2 −∞ 2π exp β p + m∗ − 1 p2 + m2∗
2
18
Semi-classical approx.
Transport equation:
!
K̂c + ! K̂q ) W = 0
2
with
K̂c Wq = −K̂q Wc + O(!2 )
K̂c Wc = 0
Wq (q, p; t) =
Thus:
!
W = Wc + ! Wq
2
dt! dq ! dp! Gc (q, q ! , p, p! ; t, t! ) K̂q Wc (q ! , p! ; t! )
Classical Green function satisfies:
∂Gc
∂H ∂Gc
∂H ∂Gc
K̂c Gc (q, q , p, p ; t, t ) ≡
+
−
= δ(t − t! ) δ(q − q ! ) δ(p − p! )
∂t
∂p ∂q
∂q ∂p
!
!
!
Solution is simply:
" !
"
!
!
Gc (q, q , p, p ; t, t ) = θ(t − t ) δ q − q̃(t) − q + q̃(t ) δ p − p̃(t) − p + p̃(t )
!
!
!
!
!
!
!
19
Non-abelian gauge field
Idea: Use generalized Gaussian wave packets to represent Wigner function in the link
representation of lattice gauge fields [Gong, BM, Biró, Nucl. Phys. A568 (1994) 727].
!
#
"
g2 " 1 a a
4
H=
El El + 4 Re
(1 − tr Up )
Lattice (KS) Hamiltonian:
a
2
g
p
l
can be scaled to dimensionless variables:
ag 2 H → H,
g 2 E → E,
t/a → t,
g2 ! → !
Wave packet ansatz:
Φ[Ul ] =
!
φl (Ul ) =
l
Continuum limit:
1
Φ [(A(x)] = √ exp
N
!
l
!"
1
√ exp
Nl
d3 x
#$
a
"
#
bl
1
−1
−1
tr(Ul Ul0 ) − tr(El0 Ul Ul0
)
2
!
%&
b(x) a
i
a
2
−
(A (x) − A0 (x)) + E0 (x)a Aa (x)
8
!
20
Time evolution
Semiclassical transport equation for Wigner functional is realized by variational
principle for wave packets. ħ then enters in two distinct ways:
! t2
[Ela , Abm ] = −i!δ ab δlm
δ
!Φ|(i! ∂t − H)|Φ# = 0
and
t1
Φ[Ul ] =
!
l
φl (Ul ) =
!
l
1
√ exp
Nl
"
#
bl
1
−1
−1
tr(Ul Ul0 ) − tr(El0 Ul Ul0
)
2
!
Variation is performed with respect to parameters: bl (t), Ul0 (t), El0 (t).
Write b = v + iw ; then express H in terms of parameters:
#
! "1
!
f (vl ) 2
2 3f (vl )
2
Heff =
(1 −
)El0 + !
|bl | + 4
(1 − fp Up0 )
2
2vl
16vl
p
l
where f (v) = I2 (2v)/I1 (2v) and fp =
!
f (vl )
l∈p
Limits: ħ→∞
vl → 0 i.e. wave packet covers all SU(2); ħ→0
vl → ∞ .
21
Eqs. of motion
dEla
dt
dUl
dt
dvl
dt
dwl
dt
=
=
=
=
i !
3! wl a
a
(fp τ Up ) −
El ,
f (vl )
8 vl
p(l)
"
#
1
i
1
−
El Ul ,
2 f (vl ) 2vl
3! f (vl ) wl
,
!
8 f (vl ) vl
"
#
"
#
2
2
!
1
1
El
2
3 2
wl
2
−
El +
+
fp Up − ! vl +
,
f (vl ) 2vl
4vl
f (vl )
16
vl
p(l)
" 2
"
##
2
wl
f (vl ) El
3 2
− !
+ ! 1− 2
,
2
f (vl ) 4vl
16
vl
ħ → 0, vl → ∞
classical limit:
!
dEla
=i
(τ a Up ) ,
dt
p(l)
Reminder:
! ↔ g 2 ! = 4παs .
dUl
= iEl Ul .
dt
22
Results - 1
Run simulation from randomly chosen initial conditions (but identical width bl) until
the system thermalizes. Measure T from electric energy distribution and determine
maximal Lyapunov exponent. Compare with classical limit. Fit gives:〈v〉≈ 7/ħ.
23
Results - 2
λ(T, !) = λc (T )(1 + δλ)
!
0.1
0.3
0.3
0.3
0.3
0.3
0.3
0.7
0.7
T
0.76
0.64
0.73
0.85
0.85
0.85
0.9
0.5
0.8
λ
0.255
0.24
0.3
0.34
0.36
0.34
0.41
0.19
0.41
δλ > 0 and grows with ħ !
δλ
0.01
0.12
0.22
0.2
0.27
0.2
0.37
0.16
0.54
!v"
70
23
22
21
21
22
20
9.0
7.4
E
2.13
2.48
2.92
3.31
3.25
3.25
3.74
3.24
4.40
24
Conclusions
Wigner functional method provides a practical method to simulate quantum
corrections to real-time evolution of fields in quantum field theory. Various
approximation methods are available:
• Gaussian wave packets (with dynamical width)
• Green function techniques
• Mean field approximation
WF method permits smooth interpolation between field eigenstates and
particle excitations. This allows for the approximate treatment of quantum
coherence and uncertainty relation effects.
WF method has been applied to SU(2) gauge field with “Gaussian” SU(2)
wave packets and variational time evolution. Results of a limited exploration
suggest that λmax grows with ħ = 4παs.
Generalization to fermions, using techniques of AMD or FMD, possible?
25